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Year 11
Higher

Problem solving with graph transformations

I can use my knowledge of graph transformations to solve problems.

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New
New
Year 11
Higher

Problem solving with graph transformations

I can use my knowledge of graph transformations to solve problems.

Download all resources
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Share activities with pupils
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Lesson details

Key learning points

  1. Your knowledge of identifying transformations of shapes can be applied to graphs
  2. This allows you to identify what transformations have taken place
  3. Once the transformations have been identified, you can write this in function notation

Keywords

  • Transformation - A transformation is a process that may change the size, orientation or position of a shape or graph.

Common misconception

Pupils may not see the link between transformations and manipulating algebraic expressions.

Be explicit in highlighting that the graph of $$y=x^2+10x+22$$ is the graph of $$y=(x+5)^2-3$$ which is the graph of $$y=x^2$$ transformed by the translation, "$$5$$ units left, $$3$$ units down".

Ask pupils questions like, "What do we know about the quadratic $$y=x^2+6x-1$$?" They should now be able to add, "It is a transformation of $$y=x^2$$ by translation, "$$3$$ units left, $$10$$ units down".
Teacher tip

Licence

This content is © Oak National Academy Limited (2024), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

Lesson video

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Starter quiz

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6 Questions

Q1.
Match the transformed functions of $$ \text {f}(x)$$ to their descriptions.
Correct Answer:$$ \text {f}(x) + 5$$,Translation of 5 in the positive $$y$$ direction

Translation of 5 in the positive $$y$$ direction

Correct Answer:$$ \text {f}(x+5) $$,Translation of 5 in the negative $$x$$ direction

Translation of 5 in the negative $$x$$ direction

Correct Answer:$$ \text {f}(5x)$$,Stretch of $$1 \over 5$$ in the $$x$$ direction

Stretch of $$1 \over 5$$ in the $$x$$ direction

Correct Answer:$$5 \text {f}(x) $$,Stretch of 5 in the $$y$$ direction

Stretch of 5 in the $$y$$ direction

Correct Answer:$$ -\text {f}(x) $$,Reflection in the $$x$$ axis

Reflection in the $$x$$ axis

Correct Answer:$$ \text {f}(-x) $$,Reflection in the $$y$$ axis

Reflection in the $$y$$ axis

Q2.
The graph of $$y= \text{f}(x)$$ is labelled. What other function of $$x$$ has been graphed?
An image in a quiz
$$ \text {f}(x + 2)$$
$$ \text {f}(x - 2)$$
Correct answer: $$ \text {f}(2x)$$
$$ 2\text {f}(x)$$
$$ \text {f}\left({1\over2}x\right)$$
Q3.
The graph of $$y= \text{f}(x)$$ is labelled. What other function of $$x$$ has been graphed?
An image in a quiz
Correct answer: $$ \text {f}(x + 2)$$
$$ \text {f}(x - 2)$$
$$ \text {f}(x) - 2$$
$$ -\text {f}(x)$$
$$ \text {f}(-x)$$
Q4.
The graph of $$y= \text{g} (x)$$ is labelled. What other function of $$x$$ has been graphed?
An image in a quiz
$$ \text {g}(x) - 8$$
$$ \text {g}(x- 8)$$
Correct answer: $$ -\text {g}(x)$$
$$ \text {g}(-x)$$
Q5.
An __________ is a line which a curve approaches but never touches.
Correct answer: asymptote
axis
exponential
image
invariant line
Q6.
Which of these is equivalent to $$x^2 - 4x + 6$$?
$$(x-2)^2-4$$
Correct answer: $$(x-2)^2+2$$
$$(x-2)^2+10$$
$$(x-4)^2-10$$
$$(x-4)^2+6$$

Exit quiz

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6 Questions

Q1.
The graph of $$y=x^2$$ has been transformed to get a new graph. What is the equation of this new graph?
An image in a quiz
$$y=x^2 +3$$
$$y=x^2 -3$$
$$y=(x+3)^2$$
Correct answer: $$y=(x-3)^2$$
Q2.
The graph of $$y=x^2$$ has been transformed to get a new graph. What is the equation of this new graph?
An image in a quiz
Correct answer: $$y=x^2 + 4x$$
$$y=x^2 - 4x$$
$$y=x^2 - 8x$$
$$y=x^2 + 4x + 8$$
$$y=x^2 + 4x - 8$$
Q3.
The graph of $$y=x^2$$ has been transformed to get a new graph. What is the equation of this new graph?
An image in a quiz
Correct answer: $$y=-4x^2$$
$$y=-2x^2$$
$$y=2x^2$$
$$y=4x^2$$
Q4.
The graph of $$y=x^2-2x$$ has been transformed to get a new graph. What is the equation of this new graph?
An image in a quiz
$$y=2x^2-2$$
$$y=2x^2+2$$
$$y=2x^2-4x+2$$
Correct answer: $$y=2x^2-8x+6$$
$$y=4x^2-8x+3$$
Q5.
The graph of $$\text{f}(x)=2^x + 1$$ has an asymptote at $$y=1$$. Match up the transformations with the equation of the asymptote after the transformation.
An image in a quiz
Correct Answer:$$\text {f}(x+3)$$,$$y=1$$

$$y=1$$

Correct Answer:$$\text {f}(x)+3$$,$$y=4$$

$$y=4$$

Correct Answer:$$3\text {f}(x)$$,$$y=3$$

$$y=3$$

Correct Answer:$$-\text {f}(x)$$,$$y=-1$$

$$y=-1$$

Q6.
The graph of $$\text{f}(x)= \text{tan}(x)$$ where $$-90 \le x \le 90$$ has asymptotes at $$x=-90$$ and $$x=90$$, which of these transformations will not change the position of the asymptotes ?
Correct answer: $$-\text{f}(x)$$
Correct answer: $$3\text{f}(x)$$
$$\text{f}(x+30)$$
Correct answer: $$\text{f}(x) + 3$$
$$\text{f}(3x)$$